Singular jets in free-falling droplets

Abstract

We report on singular jets in a free-falling liquid tin droplet following nanosecond laser-pulse impact. Following impact, the droplet (with diameter $D_0=50$ or 70\,$μ$m) undergoes rapid radial expansion and subsequent retraction, resulting in the formation of an axisymmetric jet. Using numerical simulations in tandem with our experiments, we reveal that a delicate interplay between radial flow and the curvature of the retracting droplet governs jet formation. The resulting dynamics is characterized using the impact Weber number, $\We$ (in the experiments $2 \lesssim \We \lesssim 16$), and a pressure width, W (typically $1 \lesssim \W \lesssim 2$), which describes the angular distribution over the droplet surface of the instantaneous pressure impulse exerted by the transient laser-produced plasma. %, within the range $0-20$. For values $\We<10$, the droplet presents a pronounced forward curvature during the retraction, leading to the formation of a cavity. The collapse of such a cavity leads to a singular jet that greatly enhances the jetting velocity up to ten times the impact propulsion velocity, an effect that narrowly peaks around $\We\sim6-8$, reminiscent of singular jets in droplet-solid impact. We identify a further sensitivity of the jet velocity enhancement on the pressure width W and capture the dynamics in a phase diagram connecting the various deformation morphologies with jet velocity.

Figures (9)

• Figure 1: Comparison of experiment and simulation. (a) Conceptual sketch depicting a Gaussian nano-second laser hitting the droplet with radius $R_0$ from the left. The resulting plasma after surface ablation is depicted with a yellow evolving area, where the dashed arrows indicate the plasma recoil pressure. (b) Examples of angular distributions of the plasma pressure on the droplet's surface for two different $\mathrm{We}$ values at fixed $\mathrm{W}$ (solid lines) and two different $\mathrm{W}$ values for fixed $\mathrm{We}$ (dashed lines). Conceptual illustration of the pressure width is indicated with the dashed arrow. (c-e) Comparison of the experimental data (black shapes) for droplet jetting with numerical simulations (colored shapes) performed at the same values of $\mathrm{We}$ over different fractions of capillary time $\tau_\mathrm{c}$; the values for $\mathrm{W}$ are estimated from the empirical scaling (see main text). The $\mathrm{We}$ increases from top to bottom. The $1^{\mathrm{st}}$ frame shows the droplet at rest; the $2^{\mathrm{nd}}$ frame shows the droplet at its maximum radial extension $\sim D_\mathrm{max}$; The $3^{\mathrm{rd}}$ frame depicts the jet emerging from the droplet after its contraction. The $4^{\mathrm{th}}-5^{\mathrm{th}}$ frames illustrate the jet evolution over time. The maximum jetting velocity is observed at $\mathrm{We}=8$. Note the presence of a cavity developed during retraction for $\mathrm{We}=8$ at $0.5\tau_\mathrm{c}$, which produces a singular jet (indicated with the arrow; see main text for discussion).
• Figure 2: Simulation results for singular jetting. (a) Variation of the dimensionless jet velocity $\hat{U}/U_\mathrm{cm}$ over $\mathrm{We}$ for experiments (green squares), and simulations (orange circles). (b) Four frames of the singular jetting before and after cavity collapse. The white arrows depict the local flow velocity, showing the bidirectional flow upon retraction. (c) Numerical frames of four examples of droplet jetting and the underlying cavity dynamics at different $\mathrm{We}$ and scaled values of $\mathrm{W}$ from the relation $\mathrm{W}\sim\mathrm{We}^{0.1}$. From left to right, several fractions of capillary times $\tau_\mathrm{c}$ are presented. From top to bottom: At $\mathrm{We}=3.2$, low jetting velocity caused by smooth cavity collapse and subsequent bubble entrapment. At $\mathrm{We}=6.0$, the combination of an increased radial flow and symmetric cavity collapse leads to the singular jet. The strongest jet is observed at $\mathrm{We}=7.5$ caused by asymmetric cavity breakdown, leading to the subsequent bubble entrapment. At $\mathrm{We}=12$ sheet-like expansion is observed, with $D_\mathrm{max}\sim2D_0$, and a slower jet is produced after retraction.
• Figure 3: Phase diagram (simulations) of morphologies and jet velocity enhancement as a function of $\mathrm{W}$ and $\mathrm{We}$. (a) Simulated phase diagram relating $\mathrm{W}$ with $\mathrm{We}$ illustrating different jetting behaviors. Five relevant regimes have been identified: droplet oscillation ("Oscillation", I), bubble entrapment ("Bubble", II), symmetric cavity collapse ("Cavity", III), asymmetric cavity collapse with subsequent bubble entrapment ("Bubble + Cavity", IV), and sheet-like expansion ("Sheet", V). Some representative frames for each case are depicted in (b). The gray line shows the empirical scaling that relates $\mathrm{W}$ and $\mathrm{We}$ under our experimental conditions (see main text).
• Figure 4: Simulation study of the cavity collapse and surface waves beyond the singular jetting regime, $\mathrm{We}>10$. (a) Time evolution of the cavity collapse (from left to right) at $\mathrm{We}=10$ and for different $\mathrm{W}$ values (top to bottom). The presence of CW is indicated with arrows at $0.26\tau_\mathrm{c}$. The converging CW induces stepwise cavity collapse during the droplet retraction, leading to three subsequent jets ($1^\mathrm{st},2^\mathrm{nd}$ and $3^\mathrm{rd}$). The number of observed waves decreases for higher $\mathrm{W}$ values, as shown for $\mathrm{W}=1.25,\,1.5\,\mathrm{and}\,1.75$. (b) Variation of the dimensionless velocity $\hat{U}/U_\mathrm{cm}$ with $\mathrm{W}$, corresponding to data in (a).
• Figure 5: Comparison of the dimensionless jetting velocity $\hat{U}/U_\mathrm{cm}$ over $\mathrm{We}$ from this study (simulations and experiments) and reported data in the literature for water droplets with $D_0=2\,\mathrm{mm}$ after orthogonal impact on hydrophobic surfacesbartolo_singular_2006 (black circles) and performed on non-wetting substrates zhang2022impact (red and blue circles). In case of droplet impact experiments, the jetting velocity is rescaled with the impact velocity of the droplet, $\hat{U}/U_0$. Note the similarity in the range of $\mathrm{We}$ for the singular jetting that lies around $\mathrm{We}\sim6-8$ for all cases. The secondary peak at $\mathrm{We}\sim12$ observed in Ref. bartolo_singular_2006 is attributed to upward migration and collapse of an entrapped bubble (see the main text).